Electron and Hole Mobilities in Bulk Hematite from Spin-Constrained Density Functional Theory

Transition metal oxide materials have attracted much attention for photoelectrochemical water splitting, but problems remain, e.g. the sluggish transport of excess charge carriers in these materials, which is not well understood. Here we use periodic, spin-constrained and gap-optimized hybrid density functional theory to uncover the nature and transport mechanism of holes and excess electrons in a widely used water splitting material, bulk-hematite (α-Fe2O3). We find that upon ionization the hole relaxes from a delocalized band state to a polaron localized on a single iron atom with localization induced by tetragonal distortion of the six surrounding iron–oxygen bonds. This distortion is responsible for sluggish hopping transport in the Fe-bilayer, characterized by an activation energy of 70 meV and a hole mobility of 0.031 cm2/(V s). By contrast, the excess electron induces a smaller distortion of the iron–oxygen bonds resulting in delocalization over two neighboring Fe units. We find that 2-site delocalization is advantageous for charge transport due to the larger spatial displacements per transfer step. As a result, the electron mobility is predicted to be a factor of 3 higher than the hole mobility, 0.098 cm2/(V s), in qualitative agreement with experimental observations. This work provides new fundamental insight into charge carrier transport in hematite with implications for its photocatalytic activity.

: Comparison of first nearest neighbour electronic coupling and reorganisation energy for the hole polaron, 2x2x1 and 4x4x1 supercells. The results shown in this work are for the 4x4x1 supercell of hematite, with a total of 480 atoms and 4800 electrons. It could be argued that the 2x2x1 supercell (120 atoms, 1200 electrons) is large enough for the first nearest neighbour interactions. Table 1 shows that this is true for the weak coupling directions, but not for the highest coupling direction for which we observe a non-negligible increase from 167 to 203 meV. We analysed the origin for this by preparing a transition state geometry in the 2x2x1 supercell that has exactly the same geometry as the one obtained for the 4x4x1 supercell. The coupling is virtually identical in both cases which means that the increase in coupling going from the 2x2x1 to the 4x4x1 supercell is due to a slightly different geometry of the transition state structure in the 4x4x1 supercell (rather than due to other finite size effects such as, e.g., polarisation effects due to polaron images). The slight differences in nuclear relaxation are also indicated by a small decrease in reorganisation energy going from 2x2x1 to 4x4x1, which we attribute to a smaller reorganisation of the first coordination shell ("inner sphere") in the larger supercell.
neighbours for the hole and second nearest neighbour for the electron polaron. This supercell should be large enough as the distance to the third nearest neighbour of the hole polaron is a factor of 3 smaller than to the distance to the closest periodic image. Remaining finite size effects could not be investigated as supercells larger than 4x4x1 are computationally unfeasible at the moment.
Supplementary Figure 1: Finite size effects of the hole polaron. Isosurface of CDFT weight function, showing that the distance between the third nearest neighbour of 5.9Å is a factor of three smaller than the distance to the periodic image of this charge (15.4Å).
Electronic coupling from 'energy-gap' and from the HOMO-LUMO energy gap on the DFT transition state Hole polaron

Calculation of average ET parameters
Boltzmann averages are used for combining the 9 possible transition states of the hole polaron, for both the reorganisation energy and the electronic coupling. These are calculated according to the following equations: motivated from the form of the activation free energy where ⟨|H ab | 2 ⟩ TS is the thermal average of the squared electronic coupling in the transition state (TS) and λ 4 is the activation free energy on the diabatic electronic states ∆A ‡ na . Table 3 shows a summary of the other possible combination methods: mean, root mean square and the Boltzmann average used in this work Hole polaron structure Table 4 shows the Hirshfeld spin moment and charge over the iron atom that the polaron localises. It can be seen that the difference in spin moment (+0.66 from neutral to hole ground state) is a good indicator of polaron formation while the change in charge (-0.02) is not. This is clearly visible in Figures 1-2, such that the change in spin density over the central iron atom is entirely positive (coloured yellow) while the change in electron density is both positive (yellow) and negative (blue).

Hopping schematic
Due to the 2-site delocalised electron polaron structure, there are only four symmetry related second nearest neighbours to which the polaron may hop (Supplementary Figure 5). This introduces a similar complication to the hole polaron, that for a single energy degenerate structure of the electron polaron the mobility is locally anisotropic. For ease of calculation, we consider the isotropic mobility for hopping to six second nearest neighbours and simply divide the final mobility by a factor of 4/6 to account for the four second nearest neighbours in the crystal.
Supplementary Figure 5: 2D hematite plane showing the four highest coupling neighbours (5.04 A) for the electron polaron (left), and three highest coupling neighbours (2.97Å) for the hole polaron (right).
We note that a modified version of the CP2K 8.1 code was used, with a complete implementation of Hirshfeld based CDFT including analytic forces. Code is available on request, and we intend to merge these changes into the current branch of CP2K in the future.